Unit 10: Day 3: Challenges Are Shaping Up…
Grade 8
Math Learning Goals
 Investigate the relationship of the areas of semi-circles drawn on the sides of a
right-angled triangle.
Materials
Assessment
Opportunities
Minds On… Whole Class  Discussion
Collect the Home Activity for assessment.
Using a sketch, reinforce the concept of the Pythagorean relationship. Stress that
the relationship is true for right-angled triangles only.
Ask: Does this relationship work with shapes other than squares drawn on the
right sides of a right-angled triangle?
Action!
Pair/Share  Investigation
Using grid paper, students draw a right-angled triangle. They construct semicircles on the legs and hypotenuse of the triangle and calculate the areas of each
semi-circle to determine the relationship the same way they did with squares on
Day 2. Students share their work with another pair and explain their reasoning.
N-agon areas.gsp
This GSP®4 sketch
can be used to
explore or
consolidate.
Review how to
determine the area
of a circle.
Reasoning & Proving/Observation/Checklist: Observe students as they explain
their reasoning.
Consolidate Whole Class  Discussion/Brainstorm
Debrief
Summarize the findings of their investigation. The sum of the area of the semi-
circles on the legs is equal to the area of the semi-circle on the hypotenuse.
Pythagorean relationship works for a right-angled triangle using squares and
semi-circles drawn on the sides.
Ask:
 What other shapes will work?
 Under what conditions will other shapes work?
Do not answer
these questions –
this is a brainstorm
only.
Students complete the After column for question 4 of the Anticipation Guide
(Day 2 BLM 10.2.1).
Exploration
Practice
Home Activity or Further Classroom Consolidation
Draw a right-angled triangle with the length of legs being whole numbers. On
each side of the triangle draw a rectangle (no squares are allowed!). Calculate the
areas of the three rectangles. Does this demonstrate the Pythagorean relationship?
Explain. Repeat with two more triangles.
TIPS4RM: Grade 8: Visualizing Geometric Relationships
06/02/2016
1
Unit 10: Day 3: Challenges Are Shaping Up (A)
Grade 8
Mathematical Process Goals
 Hypothesize and perform multiple trials, and draw conclusions about the
relationship among the areas of figures drawn on the sides of a right triangle.
Materials
 GSP®4
 calculators
Assessment
Opportunities
Minds On… Whole Class  Discussion
Using a sketch, reinforce the concept of the Pythagorean relationship with
squares drawn on the sides of a right angle triangle. Stress that the relationship is
true for right-angled triangles only.
Ask students to make a hypothesis about the relationship of the areas of a figure
other than a square drawn on the sides of a right-angled triangle.
Action!
Pair/Share  Investigation
Using grid paper, pairs of students draw a right-angled triangle. They construct
semi-circles on the legs and hypotenuse of the triangle and calculate the areas of
each semi-circle to determine the relationship (the same way they did with
squares on TIPS4RM Unit 10 Day 2). Students share their work with another
pair and explain their reasoning.
Mathematical Process/Reasoning and Proving/Checklist: Observe students as
they explain their reasoning.
Whole Class  Discussion
Lead a discussion to facilitate students’ understanding that since all pairs have
the same result for the investigation, it shows that it works but it doesn’t “prove”
that it is always true. It does make a convincing argument.
Mathematical
Process Focus:
Reasoning and
Proving
See TIPS4RM
Mathematical
Processes package
pp. 3–4.
Possible guiding
questions:
 Can we show that
this is true for all
cases?
 How can we
present an
argument in a
logical and
organized manner?
 What other
situations need to
be considered?
Consolidate Whole Class  Discussion/Brainstorm
Debrief
After students summarize the findings of their investigation, ask:



What other shapes do you think will work?
Under what conditions will other shapes work?
How can you show this or disprove this?
Using GSP®4, demonstrate that regardless of the length of the sides of the rightangled triangle, if the figures drawn on the sides are similar, the sum of the areas
of the figures drawn on the two legs is equal to the area of the figure drawn on
the hypotenuse.
Students draw a right-angled triangle with the length of legs being whole
numbers. On each side of the triangle they draw a rectangle (No squares are
allowed!). Calculate the areas of the three rectangles.
Ask: Does this demonstrate the Pythagorean relationship? Explain. Students
repeat with two more triangles.
Concept
Practice
This will demonstrate
a “counter-example”
and reinforce the
need for the figures
to be similar.
Home Activity or Further Classroom Consolidation
Do A or B:
A) Look around your home or neighbourhood and identify where you see rightangled triangles and show the Pythagorean relationship on it, using words,
symbols, and diagrams.
B) Considering today’s discussion make a hypothesis, about: The relationship
of the longest side of any right-angled triangle and its opposite angle. Then
try to show that it works or disprove it
Grade 8 Unit 6 Adjusted Lesson: Mathematical Processes – Representing
06/02/2016
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